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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Graph the curve and find the area that it encloses.

13. r = 2 + sin 4θ

To determine

To find: The area of the region that the polar equation encloses.

Explanation

Given:

The polar equation is r=2+sin4θ .

Assume the value of angle θ=180° .

Calculate the value of r.

r=2+sin4θ=2+sin4(180×π180)=2

Calculate the value of x coordinate of curve.

x=rcosθ

Substitute 2 for r and 180° for θ .

x=rcosθ=2cos(180×π180)=2

Calculate the value of y coordinate of curve.

y=rsinθ

Substitute 2 for r and 180° for θ .

y=2sin(180×π180)=0

Repeat the calculation of the values of x and y using the value of θ from 180° to 180° .

Tabulate the values of x and y in Table (1).

θ r=2+sin4θ x=rcosθ y=rsinθ
−180 2 −2 0
−170 2.642788 −2.60264 −0.45892
−160 2.984808 −2.8048 −1.02086
−150 2.866025 −2.48205 −1.43301
−140 2.34202 −1.79409 −1.50542
−130 1.65798 −1.06573 −1.27009
−120 1.133975 −0.56699 −0.98205
−110 1.015192 −0.34722 −0.95397
−100 1.357212 −0.23568 −1.33659
−90 2 1.23E-16 −2
−80 2.642788 0.458915 −2.60264
−70 2.984808 1.020864 −2.8048
−60 2.866025 1.433013 −2.48205
−50 2.34202 1.505422 −1.79409
−40 1.65798 1.270086 −1.06573
−30 1.133975 0.982051 −0.56699
−20 1.015192 0.953969 −0.34722
−10 1.357212 1.336593 −0.23568
0 2 2 0
10 2.642788 2.602638 0.458915
20 2.984808 2.804802 1.02084
30 2.866025 2.482051 1.433013
40 2.34202 1.794092 1.505422
50 1.65798 1.065729 1.270086
60 1.133975 0.566987 0.982051
70 1.015192 0.347216 0.953969
80 1.357212 0.235677 1.336593
90 2 1.23E-16 2
100 2.642788 −0.45892 2.602638
110 2.984808 −1.02086 2.804802
120 2.866025 −1.43301 2.482051
130 2.34202 −1.50542 1.794092
140 1.65798 −1.27009 1.065729
150 1.133975 −0.98205 0.566987
160 1.015192 −0.95397 0.347216
170 1.357212 −1.33659 0.235677
180 2 −2 0

Plot the graph as below.

From the graph, it is observed that the curve lies on the entire quadrant. Thus, the limit tends to be between 0θ2π .

Calculate the area of the region using the polar area formula

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